1. Introduction to Bayesian inference
2. The BayesFactor package
3. \(t\)-Test
4. ANOVA
5. Linear model setup

• Effect of media stories on attitudes towards Syrian refugees.
• IV: Sympathetic media story about refugees or control stofy about food trends.
• DV: Participants’ favorability ratings towards refugees.
• Also recorded: Political affiliation of the participants.

Example from Rouder, Haaf, & Aust (2018)

• Conduct a \(t\)-test to assess the effect of media stories on attitudes towards Syrian refugees.
• Conduct a 2x2 ANOVA to assess the effects of both media stories and political affiliation as well as their interaction.
• Neither necessarily match the research questions associated with the study.
• Baysian model comparison allows to better match inference and theoretical questions.
• But you can also do the regular stuff.

1. The plight of refugees generates an empathic response which results in an increase in favorability for conservative and liberals alike.
2. Confirmation bias (Tversky & Kahneman, 1974): People ignore media stories incompatible with their world view. Therefore, only liberals are affected.
3. Belief polarization (Cook & Lewandowsky, 2016): Liberals see refugees more favorably after viewing positive stories while conservatives might actually see them more negatively.

• We need models

• We need models
• \({\cal M}_1:\) The effect of media stories on attitudes is positive.

• We need models
• \({\cal M}_1:\) The effect of media stories on attitudes is positive.
• \({\cal M}_0:\) There is no effect of media stories on attitudes.

• The prior distribution indicates the plausibility of effects before the data have been collected.
• Because we make this fine-grained specification, predictions are now possible.

• In Bayesian statistics, evidence for models reflects how well they predict data.
• Prior is on true values.
• Predictions are on data.

• In Bayesian statistics, evidence for models reflects how well they predict data.

• In Bayesian statistics, evidence for models reflects how well they predict data.

• In Bayesian statistics, evidence for models reflects how well they predict data.

• In Bayesian statistics, evidence for models reflects how well they predict data.

• Based on Rouder, Speckman, Sun, Morey, & Iverson (2009) (\(t\)-test), Rouder et al. (2012) (ANOVA), and Rouder & Morey (2012) (regression), total of > 4000 citations.
• Same setup is implemented in JASP and SPSS.
• Provides a convenient interface for commonly used models.
• Bayesian versions of typical frequentist tests using Bayes factor.

• General linear models (including linear mixed effects models): generalTestBF, lmBF
• Linear regression: regressionBF, lmBF
• Linear correlation: correlationBF
• \(t\)-tests: ttestBF
• Meta-analytic \(t\)-tests: meta.ttestBF
• ANOVA: anovaBF, lmBF
• Contingency tables: contingencyTableBF
• Single proportions: proportionBF

• General linear models (including linear mixed effects models): generalTestBF, lmBF
• Linear regression: regressionBF, lmBF
• Linear correlation: correlationBF
• \(t\)-tests: ttestBF
• Meta-analytic \(t\)-tests: meta.ttestBF
• ANOVA: anovaBF, lmBF
• Contingency tables: contingencyTableBF
• Single proportions: proportionBF

Can be used to test our position 1 .

Equation setup

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study)

• \({\cal M}_1:\) The effect of media stories on attitudes is positive.
• \({\cal M}_0:\) There is no effect of media stories on attitudes. >- Attitudes might be normally distributed: \(Y_{ij} \sim \mbox{Normal}(\mu + x_j \theta, \sigma^2)\). >- For model \({\cal M}_0\) \(\theta = 0\). >- For model \({\cal M}_1\) we need a prior on mu: \(\theta\) >- Prior on \(\sigma^2\) and \(\mu\)?

Obtain strong evidence by increasing model discriminability.

Stefan, Gronau, Schönbrodt, & Wagenmakers (2019)

• Complex models often suffer from identifiability issues.

• Informed priors introduce probabilistic constraints.

• BayesFactor package in R and JASP use default priors for the t-test called JZS prior.
• Prior structure is consistent.
• User-defined prior settings are on the scale of the effect size:
• \(\delta = \theta / \sigma\) (think Cohen’s \(d\)).

• BayesFactor package in R and JASP use default priors for the t-test called JZS prior.
• Prior structure is consistent.
• User-defined prior settings are on the scale of the effect size.

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study
                     , rscale = 1 / sqrt(2))

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study
                     , rscale = 1 / sqrt(2))

par(mar = c(3,3,.5,.5), mgp = c(2,.7,0))
x <- seq(-3, 3, .01)
y <- dcauchy(x, 0, 1/sqrt(2))
plot(x, y, type = "l", lwd = 2
     , ylab = "Density", xlab = "Effect Size"
     , col = "darkblue", ylim = c(0, .5))
lines(x, dnorm(x), col = adjustcolor(1, .5), lty = 2, lwd = 2)
abline(v = c(-1, 0, 1), lwd = 0.5, col = adjustcolor(1, alpha.f = 0.3))

##         0%        25%        50%        75%       100% 
## -755409.12      -1.24      -0.01       1.21  109714.01

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study
                     , rscale = 1 / 4)

1. Plot the default prior on effect size.
2. Adjust the scale until it matches your expectations of the outcome of the study.
3. Conduct a \(t\)-test using your setting.
4. Do you think any two people in the workshop get the same result?

Put your setting in the zoom chat!

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study
                     , rscale = 1 / 2)

The plight of refugees generates an empathic response which results in an increase in favorability for conservative and liberals alike.

• Does the \(t\)-test as we used it address this statement?

est.attitude <- BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study, rscale = 1 / 2
                     , posterior = T, iterations = 20000)
plot(est.attitude[,"beta (Neutral - Refugee plight)"])

The plight of refugees generates an empathic response which results in an increase in favorability for conservative and liberals alike.

• Does the \(t\)-test as we used it address this statement?

• We need an ordinal constraint.
• Ordinal constraint: Specification of the direction of an effect.

• One-sided \(t\)-tests are easy with BayesFactor.

• One-sided \(t\)-tests are easy with BayesFactor.

BayesFactor::ttestBF(formula = attitude ~ story
                     , data = immigration_study
                     , rscale = 1 / 2
                     , nullInterval = c(-Inf, 0))

## Bayes factor analysis
## --------------
## [1] Alt., r=0.5 -Inf<d<0    : 4.196947   ±0%
## [2] Alt., r=0.5 !(-Inf<d<0) : 0.08881904 ±0%
## 
## Against denominator:
##   Null, mu1-mu2 = 0 
## ---
## Bayes factor type: BFindepSample, JZS

1. New data set, this time real data.
2. Run a Bayesian t-test, try out ordinal constraints and estimation, maybe even prior prediction.
3. As questions while I am here!

SourceURL <- "https://raw.githubusercontent.com/PerceptionCognitionLab/data0/master
/contexteffects/FlankerStroopSimon/cleaning.R"
devtools::source_url(SourceURL)

library(tidyverse)

stroop.agg <- stroop %>%
  group_by(ID, congruency) %>%
  summarize(mrt = mean(RT)) %>%
  spread(congruency, mrt)

• Based on Rouder et al. (2012).
• Between-subject, within-subject and mixed designs.
• Random effects.
• Perhaps the most thought out functionlity in BayesFactor.